Post on 17-Mar-2022
transcript
TRAVERSAL OF DIRECTED GRAPHS
Partha P Chakrabarti Indian Institute of Technology Kharagpur
Directed GraphsAn Undirected Graph G = (V, E) consists of
the following:
• A set of Vertices or Nodes V
• A set of DIRECTED Edges E where each
edge connects two vertices of V. The
edge is an ORDERED pair of vertices
Successor Function: succ(i) = {set of nodes
to which node i is connected}
Directed Acyclic Graphs (DAGs): Such
Graphs have no cycles (Figure 2)
Weighted Undirected Graphs: Such Graphs
may have weights on edges (Figure 3). We
can also have Weighted DAGs
Figure 1
Figure 3Figure 2
Basic Traversal Algorithm (Depth First Search)
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL /
succ(i) = {set of nodes to which node i is connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
}
}
Traversing the Complete Graph by DFS
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL /
succ(i) = {set of nodes to which node i is connected}
Dfs(node) {
visited[node] = 1;
for each j in succ(node) do {
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
}
}
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected}
numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Entry-Exit Numbering
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected} numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Tree Edge, Back Edge, Forward Edge, Cross Edge
Edge (u,v) is
Tree Edge or Forward Edge: if & only if
Entry[u] < Entry[v] < Exit[v] < Exit[u]
Back Edge: if & only if
Entry[v] < Entry [u] < Exit [u] < Exit [v]
Cross Edge: if & only if
Entry [v] < Exit [v] < Entry [u] < Exit [u]
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected} numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Reachability, Paths, Cycles, Components
Edge (u,v) is
Tree Edge or Forward Edge: if & only if
Entry[u] < Entry[v] < Exit[v] < Exit[u]
Back Edge: if & only if
Entry[v] < Entry [u] < Exit [u] < Exit [v]
Cross Edge: if & only if
Entry [v] < Exit [v] < Entry [u] < Exit [u]
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected} numb = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb = numb + 1;
Exit[node] = numb;
}
Directed Acyclic Graphs
Global Data: G = (V,E)
visited [i] indicates if node i is visited. / initially 0 /
Parent[i] = parent of a node in the Search / initially NULL /
Entry[i] = node entry sequence / initially 0 /
Exit[i] = node exit sequence / initially 0 /
succ(i) = {set of nodes to which node i is connected} numb = 0; numb1 = 0;
Dfs(node) {
visited[node] = 1; numb = numb+1;
Entry[node] = numb;
for each j in succ(node) do
if (visited [j] ==0) { Parent[j] = node;
Dfs(j) }
numb1 = numb1 + 1;
Exit[node] = numb1;
}
Topological Ordering, Level Values
Shortest Cost Path in Weighted DAGs
Breadth-First SearchGlobal Data: G = (V,E)
Visited[i] all initialized to 0
Queue Q initially {}
BFS(k) {
visited [k] = 0; Q = {k};
While Q != {} {
j = DeQueue (Q);
if visited[j] == 0 {
visited [j] = 1;
For each k in succ (j) {
if (visited[k]==0) EnQueue(Q,k); }
}
}
/Parent links, Shortest Length Path Finding in unweighted directed graphs/
Pathfinding in Weighted Directed Graphs Global Data: G = (V,E)
Visited[i] all initialized to 0,
Cost[j] all initialized to INFINITY
Ordered Queue Q initially {}
BFSW(k) {
visited [k] = 0; cost [k] = 0; Q = {k};
While Q != {} {
j = DeQueue (Q);
if visited[j] == 0 {
visited [j] = 1;
For each k in succ (j) {
if cost[k] > cost[j] + c[j,k]
cost[k] = cost[j] + c[j,k];
EnQueue(Q,k);}
}
}
Thank you